Probability — Class 10 Mathematics
"Probability: the measure of how likely something is to happen — a number between 0 (impossible) and 1 (certain)."
1. About the Chapter
Probability quantifies UNCERTAINTY. It's used in:
- Insurance (calculating risks)
- Weather forecasting
- Medicine (treatment effectiveness)
- Sports analytics
- Computer science (machine learning)
- Cricket strike rates
Class 10 Focus
Mostly THEORETICAL probability (assuming equally likely outcomes).
Closes Class 10 Math
This is the final chapter of Class 10. Brief, but important foundation.
2. Basic Definitions
Random Experiment
An experiment whose outcome cannot be predicted with certainty.
- Tossing a coin
- Rolling a die
- Drawing a card from deck
Outcome
A possible result of an experiment.
- Tossing coin: Head (H) or Tail (T)
- Rolling die: 1, 2, 3, 4, 5, or 6
Sample Space (S)
The set of ALL possible outcomes.
- Tossing 1 coin: S = {H, T}
- Rolling 1 die: S = {1, 2, 3, 4, 5, 6}
Event
A subset of the sample space.
- 'Getting a head' = {H}
- 'Getting an even number' = {2, 4, 6}
Favourable Outcomes
Outcomes in which the desired event occurs.
3. Probability Formula
Classical Probability
P(E) = (Number of favourable outcomes) / (Total number of outcomes)
= n(E) / n(S)
Range
0 ≤ P(E) ≤ 1
- P(E) = 0: IMPOSSIBLE event
- P(E) = 1: SURE/CERTAIN event
- 0 < P(E) < 1: Possible
Sum of Probabilities
For all outcomes in sample space: SUM = 1.
For an event E and its complement E': P(E) + P(E') = 1
So: P(E') = 1 − P(E)
4. Common Examples
Coin Toss
- 1 coin: S = {H, T}; P(H) = 1/2
- 2 coins: S = {HH, HT, TH, TT}; P(at least one H) = 3/4
Dice
- 1 die: S = {1, ..., 6}
- P(prime) = 3/6 = 1/2 (primes: 2, 3, 5)
- P(even) = 3/6 = 1/2
- P(> 4) = 2/6 = 1/3
Two Dice
- Total outcomes: 6 × 6 = 36
- Possible sums: 2 to 12
- P(sum = 7) = 6/36 = 1/6 (highest probability!)
Cards (52 in a deck)
- 4 suits: Hearts, Diamonds (red); Clubs, Spades (black)
- 13 cards per suit: A, 2, 3, ..., 10, J, Q, K
- Face cards: J, Q, K (12 total: 3 × 4)
- P(any specific card) = 1/52
- P(ace) = 4/52 = 1/13
- P(face card) = 12/52 = 3/13
- P(red card) = 26/52 = 1/2
- P(spade) = 13/52 = 1/4
5. Complementary Events
Definition
For event E, the complementary event E' (or NOT E) contains all outcomes NOT in E.
Formula
P(E) + P(NOT E) = 1
Example
If P(passing exam) = 0.8, then P(failing) = 1 − 0.8 = 0.2.
Useful When
- Direct counting is hard
- 'At least one' problems
6. Worked Examples
Example 1: Single Die
A die is rolled. Find P(getting a number < 4).
- Favourable: 1, 2, 3 (three outcomes)
- Total: 6
- P = 3/6 = 1/2
Example 2: Card Draw
A card is drawn from 52-deck. Find: (a) P(red card) = 26/52 = 1/2 (b) P(king) = 4/52 = 1/13 (c) P(red king) = 2/52 = 1/26 (d) P(card with letter on it) = 16/52 = 4/13 (Ace + J + Q + K)
Example 3: Two Coins
Two coins are tossed. Find P(exactly one head).
- Sample space: {HH, HT, TH, TT}
- Favourable: {HT, TH}
- P = 2/4 = 1/2
Example 4: Box of Balls
A box contains 3 red, 5 blue, 2 green balls. A ball is drawn at random. (a) P(red) = 3/10 (b) P(blue) = 5/10 = 1/2 (c) P(not red) = 7/10 (d) P(red or blue) = 8/10 = 4/5
Example 5: Two Dice — Sum
Two dice rolled. Find P(sum = 8).
- Total outcomes: 36
- Favourable: (2,6), (3,5), (4,4), (5,3), (6,2) — 5 outcomes
- P = 5/36
Example 6: Birthday Problem
A class has 30 students. Probability that 2 have same birthday?
- Very HIGH (≈ 70%) — famous birthday paradox
- (Beyond Class 10 syllabus but interesting)
7. Geometric Probability (Brief)
When outcomes lie in a geometric region: P = (Favourable region area) / (Total region area)
Example
A point is chosen randomly inside a square of side 2. Probability it lies in inscribed circle?
- Circle area = π(1)² = π
- Square area = 4
- P = π/4 ≈ 0.785
8. Common Mistakes
-
Forgetting all outcomes equally likely
- Classical probability ASSUMES equal likelihood. For weighted dice, biased coins, this fails.
-
Wrong sample space
- For 2 coins, S = {HH, HT, TH, TT} (4 outcomes), not 3 (HH, HT, TT).
-
Adding probabilities of intersecting events
- P(A or B) ≠ P(A) + P(B) if events overlap. (Need inclusion-exclusion, Class 11.)
-
Confusing P(E) and P(E')
- P(E) + P(E') = 1, so P(NOT E) = 1 − P(E).
-
Probability > 1
- IMPOSSIBLE. If you get P > 1, you've made an error.
9. Real-World Applications
Insurance
Companies calculate probability of accidents, deaths to set premiums.
Weather
'70% chance of rain' = probability statement.
Cricket
- Run rate, strike rate use probability concepts
- Probability of winning given current score
Medicine
- Probability of recovering from a disease
- Drug effectiveness in trials
Gambling/Games
- Lottery: P(winning) ≈ 1/14,000,000 (extremely low!)
- Poker probabilities
Indian Context
- IPL match predictions use probability
- Election polling
- Indian Statistical Institute leads probability research
10. Famous Probability Problems
Coin Toss Paradox
If you toss a fair coin 10 times and get 10 heads in a row, P(11th toss being head) = ?
- Still 1/2!
- Each toss is INDEPENDENT.
Monty Hall Problem
3 doors, prize behind one. You pick door 1. Host opens door 3 (empty). Should you switch to door 2?
- YES! Switching increases probability from 1/3 to 2/3.
- Famous and counterintuitive.
Birthday Paradox
In a room of 23 people, probability that 2 share a birthday is over 50%.
- Counterintuitive but true.
11. Conclusion
Probability is the mathematics of UNCERTAINTY:
- We can quantify how likely things are
- Make informed decisions under uncertainty
- Foundation for statistics, machine learning, gambling theory
Master:
- Formula: P = favourable / total
- Sample space for various experiments
- Complementary events
- Word problems (cards, dice, coins, real-life)
This chapter is BRIEF but IMPORTANT for board exam (~6-8 marks).
In Class 11-12, you'll learn:
- Conditional probability
- Bayes' theorem
- Probability distributions
- Statistics with probability
For now, master the basics. Practice 15+ problems.
Probability: the math of guessing right.
